In the previous chapter we considered the idealized case of a cold plasma in which there were no random thermal motions. In this chapter we introduce a more general framework for analyzing plasmas in which thermal motions must be considered. For a system with a large number of particles it is neither possible nor desirable to determine the motion of each and every particle. Instead, we will use a statistical approach to compute the average motion of a large number of particles. This approach is called kinetic theory. It is not our intention in this chapter to present an exhaustive treatment of plasma kinetic theory. Instead we will introduce the basic concepts of kinetic theory and derive a system of equations known as the moment equations. These equations will then be used to analyze various simple applications that are of interest.

The distribution function

To carry out a statistical description of a plasma, it is convenient to introduce a six-dimensional space, called phase space, that consists of the position coordinates x, y, and z, and the velocity coordinates vx, vy, and vz. At any given time the dynamical state of a particle can be represented by a point in phase space. For a system of many particles, the dynamical state of the entire system can then be represented by a collection of points in phase space, with one point for each particle.